after you have gone to the exchange page, while standing on the planet surface (not in a ship or vehicle, building of any sort) you will see a list of ships and ehicles at the bottom of the page.

Some will be in red (you cant afford them) and some will be in green (you can afford one). Next to the names will be a box saying quantity so just pop in a 1 or however many you want and click exchange.

The ship/vehicle you wanted will appear at your location and you can fly off in it.